Another stream of work (including the present paper) introduces alternative risk criteria other
than the mean-variance objective, so as to formulate the problem into a more tractable form. For
example, [1]} formulate the problem as an expected utility maximization problem, and
derive a Hamilton-Jacobi-Bellman (HJB) equation that characterizes the optimal adaptive
strategy. They find that the optimal strategy is either aggressive- or passive-in-the-money,
respectively, if and only if the utility function displays increasing or decreasing risk aversion,
respectively, but an analytic solution is not available. [2]} propose an
alternative risk criterion that utilizes the time-averaged risk exposure to the price change
driven by the geometric Brownian motion (more precisely, the risk term is formulated as the time
integration of the position value process, i.e., the product of the position process and the price
process), and explicitly solve for the optimal strategy that is shown to exhibit the AIM
behavior. However, their proposed risk criteria is ad hoc, and may encourage, for example,
negative positions to reduce risk. Hence, the resulting optimal policies are not
“liquidate-only”: they may trade in both directions (buying and selling) in excess of what
is strictly necessary to reduce the position to zero. [3]} investigate the use of the
quadratic variation of the position value process as a risk criteria, and observe that the classic
static solution of [4]} is again optimal when the price process is driven by the
arithmetic Brownian motion. The authors also consider the geometric Brownian motion, under which
the optimal solution is numerically determined and shown to be price-adaptive, and they report its
AIM behavior through numerical examples. [5]}
introduces a composite dynamic coherent risk measures and derive the optimal solution that is
tractable but static. One can also consider an entropic risk measure introduced such as that of
[6]}, but it can be shown that the resulting strategy is also not price-adaptive.
